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From Wikipedia:

In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

I assume that this theorem doesn't apply to 3D objects, because each "region" could be touching every other region. Is there any kind of related theorem that applies to 3D objects?

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    $\begingroup$ The one about a bunch of spheres in $\mathbb R^3$ that don't intersect where you form a graph by connecting two spheres if they touch? $\endgroup$ – Will Sawin Oct 7 '15 at 0:57
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    $\begingroup$ Have you looked through this older thread on what seems to be the same general question: mathoverflow.net/questions/189097/… $\endgroup$ – ARupinski Oct 7 '15 at 1:11
  • $\begingroup$ I vaguely remember a torus can be colored with seven colors and maybe an analogous result for the 3 dimensional space, but I read that some 25 years ago so take it with a grain of salt. $\endgroup$ – Sylvain JULIEN Mar 29 at 15:12
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This is an easy result, not at all comparable to the $4$-color theorem, but it perhaps has the flavor you are seeking:

A collection of tetrahedra forming a pure simplicial complex may be "solid 4-colored" so that no two glued face-to-face receive the same color. (arXiv abstract.)

The planar version is that any analogous collection of triangles can be $3$-colored.


         


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